Suppose we have an (not necessarily convex) optimization problem :
Let . Then the above problem can be equivalently written as:
The dual of the above problem can be written as:
We say that strong duality holds at a point when
By weak duality, the inequality
always holds true. My doubt is: suppose there exists an that minimizes for a fixed , can i say that the following inequality holds true:
If the above is true, then are we saying that if there is a primal variable that attains its minimum in the dual problem, then strong duality holds true? Somewhere this does not seem to add up.